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Mirrors > Home > MPE Home > Th. List > volf | Structured version Visualization version GIF version |
Description: The domain and codomain of the Lebesgue measure function. (Contributed by Mario Carneiro, 19-Mar-2014.) |
Ref | Expression |
---|---|
volf | ⊢ vol:dom vol⟶(0[,]+∞) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ovolf 24862 | . . . . . 6 ⊢ vol*:𝒫 ℝ⟶(0[,]+∞) | |
2 | ffun 6676 | . . . . . 6 ⊢ (vol*:𝒫 ℝ⟶(0[,]+∞) → Fun vol*) | |
3 | funres 6548 | . . . . . 6 ⊢ (Fun vol* → Fun (vol* ↾ dom vol)) | |
4 | 1, 2, 3 | mp2b 10 | . . . . 5 ⊢ Fun (vol* ↾ dom vol) |
5 | volres 24908 | . . . . . 6 ⊢ vol = (vol* ↾ dom vol) | |
6 | 5 | funeqi 6527 | . . . . 5 ⊢ (Fun vol ↔ Fun (vol* ↾ dom vol)) |
7 | 4, 6 | mpbir 230 | . . . 4 ⊢ Fun vol |
8 | resss 5967 | . . . . . 6 ⊢ (vol* ↾ dom vol) ⊆ vol* | |
9 | 5, 8 | eqsstri 3983 | . . . . 5 ⊢ vol ⊆ vol* |
10 | fssxp 6701 | . . . . . 6 ⊢ (vol*:𝒫 ℝ⟶(0[,]+∞) → vol* ⊆ (𝒫 ℝ × (0[,]+∞))) | |
11 | 1, 10 | ax-mp 5 | . . . . 5 ⊢ vol* ⊆ (𝒫 ℝ × (0[,]+∞)) |
12 | 9, 11 | sstri 3958 | . . . 4 ⊢ vol ⊆ (𝒫 ℝ × (0[,]+∞)) |
13 | 7, 12 | pm3.2i 472 | . . 3 ⊢ (Fun vol ∧ vol ⊆ (𝒫 ℝ × (0[,]+∞))) |
14 | funssxp 6702 | . . 3 ⊢ ((Fun vol ∧ vol ⊆ (𝒫 ℝ × (0[,]+∞))) ↔ (vol:dom vol⟶(0[,]+∞) ∧ dom vol ⊆ 𝒫 ℝ)) | |
15 | 13, 14 | mpbi 229 | . 2 ⊢ (vol:dom vol⟶(0[,]+∞) ∧ dom vol ⊆ 𝒫 ℝ) |
16 | 15 | simpli 485 | 1 ⊢ vol:dom vol⟶(0[,]+∞) |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 397 ⊆ wss 3915 𝒫 cpw 4565 × cxp 5636 dom cdm 5638 ↾ cres 5640 Fun wfun 6495 ⟶wf 6497 (class class class)co 7362 ℝcr 11057 0cc0 11058 +∞cpnf 11193 [,]cicc 13274 vol*covol 24842 volcvol 24843 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2708 ax-sep 5261 ax-nul 5268 ax-pow 5325 ax-pr 5389 ax-un 7677 ax-cnex 11114 ax-resscn 11115 ax-1cn 11116 ax-icn 11117 ax-addcl 11118 ax-addrcl 11119 ax-mulcl 11120 ax-mulrcl 11121 ax-mulcom 11122 ax-addass 11123 ax-mulass 11124 ax-distr 11125 ax-i2m1 11126 ax-1ne0 11127 ax-1rid 11128 ax-rnegex 11129 ax-rrecex 11130 ax-cnre 11131 ax-pre-lttri 11132 ax-pre-lttrn 11133 ax-pre-ltadd 11134 ax-pre-mulgt0 11135 ax-pre-sup 11136 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2890 df-ne 2945 df-nel 3051 df-ral 3066 df-rex 3075 df-rmo 3356 df-reu 3357 df-rab 3411 df-v 3450 df-sbc 3745 df-csb 3861 df-dif 3918 df-un 3920 df-in 3922 df-ss 3932 df-pss 3934 df-nul 4288 df-if 4492 df-pw 4567 df-sn 4592 df-pr 4594 df-op 4598 df-uni 4871 df-iun 4961 df-br 5111 df-opab 5173 df-mpt 5194 df-tr 5228 df-id 5536 df-eprel 5542 df-po 5550 df-so 5551 df-fr 5593 df-we 5595 df-xp 5644 df-rel 5645 df-cnv 5646 df-co 5647 df-dm 5648 df-rn 5649 df-res 5650 df-ima 5651 df-pred 6258 df-ord 6325 df-on 6326 df-lim 6327 df-suc 6328 df-iota 6453 df-fun 6503 df-fn 6504 df-f 6505 df-f1 6506 df-fo 6507 df-f1o 6508 df-fv 6509 df-riota 7318 df-ov 7365 df-oprab 7366 df-mpo 7367 df-om 7808 df-1st 7926 df-2nd 7927 df-frecs 8217 df-wrecs 8248 df-recs 8322 df-rdg 8361 df-er 8655 df-map 8774 df-en 8891 df-dom 8892 df-sdom 8893 df-sup 9385 df-inf 9386 df-pnf 11198 df-mnf 11199 df-xr 11200 df-ltxr 11201 df-le 11202 df-sub 11394 df-neg 11395 df-div 11820 df-nn 12161 df-2 12223 df-3 12224 df-n0 12421 df-z 12507 df-uz 12771 df-rp 12923 df-ico 13277 df-icc 13278 df-fz 13432 df-seq 13914 df-exp 13975 df-cj 14991 df-re 14992 df-im 14993 df-sqrt 15127 df-abs 15128 df-ovol 24844 df-vol 24845 |
This theorem is referenced by: volsup 24936 volsup2 24985 volivth 24987 itg1climres 25095 itg2const2 25122 itg2gt0 25141 areambl 26324 voliune 32868 volfiniune 32869 volmeas 32870 volsupnfl 36152 areacirc 36200 arearect 41578 areaquad 41579 volioof 44302 volicoff 44310 voliooicof 44311 fourierdlem87 44508 voliunsge0lem 44787 volmea 44789 hoidmv1lelem1 44906 hoidmv1lelem2 44907 hoidmv1lelem3 44908 ovolval4lem1 44964 ovolval5lem1 44967 |
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